Submersions, fibrations and bundles

Abstract

When does a submersion have the homotopy lifting property? When is it a locally trivial fibre bundle? We establish characterizations in terms of consistency in the topology of the neighbouring fibres. In differential topology, one meets nonproper submersive maps, and hopes that they will be fibrations (resp. fibre bundles) under hypotheses of consistency between the homotopy type (resp. topology) of the neighbouring fibres. The aim of this paper is to give suitable characterizations. I. Submersions and Fibrations This first part of this paper belongs to the most elementary homotopy theory. Our purpose is to establish the following homotopy lifting characterization, and a few corollaries. Theorem A. A surjective map is a fibration if and only if it satisfies the following three conditions: it is a homotopic submersion, all vanishing cycles of all dimen-sions are trivial, and all emerging cycles of all dimensions are trivial. Let us first specify definitions, conventions and notations. I.1. Definitions. Throughout this paper, " space " means Hausdorff topological space, " map " means continuous map, " polytope " means finite simplicial complex. For every p ≥ 0, denote by B p the compact p-ball and S p = ∂B p+1 the p-sphere. Fix a basepoint * ∈ S p . Let E, B be two spaces and π : E → B a map. As usual, by a homotopy for the map f : X → Y , we mean a map F : X ×[0, 1] → Y such that F (x, 0) = f (x) for every x ∈ X ; and we call (E, π), or π, a fibration, or equivalently we say that it has the homotopy lifting property, if for every map f : X → E whose source X is a polytope, every homotopy for π • f lifts to a homotopy for f . More generally, call π an r-fibration if this is true for every polytope X of dimension at most r . A 0-fibration is also said to have the path lifting property. Here is another generalization of fibrations. Two homotopies F, F : X × [0, 1] → Y